LCM AND HCF: Definition and Methods – MINDSPARK.

LCM AND HCF DEFINITION

LCM, i.e, the least common multiple of two or more numbers is the smallest number which is a multiple of all the numbers of which we have to find the LCM. Let us take two numbers 10 and 15.

Multiples of 10: 10, 20, 30, 40, 50, 60, …..

Multiples of 15: 15, 30, 45, 60, 75, ….

The common multiples of 10 and 15 are 30, 60, … and so on.

Therefore the least common multiple (LCM) of 10 & 15 is 30.

We also write this as LCM(10, 15) = 30.

HCF i.e, the highest common factor (also known as the greatest common factor (GCF) of two or more numbers is the largest number which divides the numbers completely of which we have to find the HCF. 

Let us take two numbers 12 and 15.

The HCF of 12 and 15 will be 3 as 3 is the greatest number which divides 12 and 15 completely.

We also write this as HCF(12, 15) = 3.

Finding LCM and HCF by 

1. PRIME FACTORISATION-

LCM – 

Step 1 –  Write the numbers in the form of their prime factors.

Step 2 – Separate the common prime factors from the factors which are not common.

Step 3 – LCM of the given numbers = product of common prime factors × product of uncommon prime factors.

Find the LCM of 12 and 16.

12 = 2 × 2 × 3 = 2^{2} \times 3

16 = 2 \times 2 \times 2 \times 2=2^{4}

So in 12 and 16, 2 × 2 are the common prime factors and 2 × 2 × 3 are the prime factors that are not common.

Therefore LCM(12,16) =2 \times 2 \times 2 \times 2 \times 3=2^{4} \times 3=48

HCF –

Step 1 –  Write the numbers in the form of their prime factors. 

Step 2 – Separate the common prime factors from the factors which are not common.

Step 3 – HCF of the given numbers = product of common prime factors.

Find the HCF of 12 and 16.

12 = 2 × 2 × 3 = 2^{2} \times 3

So in 12 and 16, 2 × 2 are the common prime factors and 2 × 2 × 3 are the prime factors that are not common

Therefore HCF(12,16) = 2 × 2 = 2² = 4

2. DIVISION METHOD – 

LCM- 

Step 1 –  Write the numbers and divide them by the smallest prime factor.

Step 2 – Write the quotient in the next line and consider it as the new dividend. Again divide it by the smallest prime factor.

Step 3 – Repeat step 2 till we get 1 as the quotient.

Step 4 – The product of all the prime factors will be the LCM.

Find the LCM of 12 and 16.

First, we take 2 as it is the smallest prime number that divides 12 and 16 completely.

Then we get 6 and 8 as the quotient. 

Now we will take 6 and 8 as the dividend.

Then again 2 is the smallest prime number which divides 6 and 8 completely.

Now we will repeat the steps till we get 1 as the quotient.

Therefore LCM(12,16) = 2 × 2 × 2 × 2 × 3 = 2^4 × 3 = 48.

We also get to know that changing the method will not change the answer.

HCF-

Step 1 –  Write the numbers of which you have to find the HCF. Divide the greater number by the smaller number and check the remainder.

Step 2 – Take the remainder as the new divisor and the old divisor as a new dividend.

Step 3 – Repeat these steps till we get 0 as the remainder.

Step 4 – The divisor which will give the remainder 0 will be the HCF of the numbers.

Find the HCF of 12 and 16.

First, we will divide 12 by 16, we will get 4 as the remainder. Now we will make 4 our new divisor and 12 our new dividend, we will get 0 as the remainder. Therefore 4 will be the HCF of 12 and 16.

HCF(12,16) = 4

LCM(a, b) × HCF(a, b) = a × b , Where a and b are Natural Numbers.

EXAMPLES

1. Find the LCM and HCF of 10 and 25.

Solution:

10 = 2 × 5

LCM(10, 25) = product of common prime factors × product of uncommon prime factors

                      = 5 × 5 × 2 = 5² × 2

                      = 50

HCF(10, 25) =  product of common prime factors

                     = 5 ( as 5 is the only common factor)

2. Find the LCM and HCF of 36 and 45.

36=3 \times 2 \times 3 \times 2=3^{2} \times 2^{2}
45=3 \times 3 \times 5=3^{2} \times 2

LCM(36,45) = product of common prime factors × product of uncommon prime factors

                     = 2 \times 2 \times 5 \times 3 \times 3=2^{2} \times 3^{2} \times 5

                     = 180

HCF(36,45) =  product of common prime factors

                    = 3 × 3  = 3²

                    = 9